President Trump will temporarily reorganize his White House staff to focus on confirming a Supreme Court nominee to fill the seat of Justice Anthony M. Kennedy.CreditCreditT.J. Kirkpatrick for The New York Times

WASHINGTON — President Trump interviewed four candidates on Monday to take Justice Anthony M. Kennedy’s place on the Supreme Court as the White House raced to meet the president’s promise to announce a replacement for the retiring justice early next week.

The White House refused to disclose the names of whom the president met with, but according to people briefed on the vetting process, they were the federal appeals court judges Amy Coney Barrett of the Seventh Circuit; Brett M. Kavanaugh of the District of Columbia Circuit; and Raymond M. Kethledge and Amul R. Thapar of the Sixth Circuit. The president met alone with them for 45 minutes each.

Judge Thapar, the son of Indian-American immigrants, was Mr. Trump’s first nominee to an appeals court in 2017. A former district court judge from Kentucky with a conservative track record, Judge Thapar was among those the president considered as a replacement for Justice Antonin Scalia, who died in 2016.

Judge Kavanaugh, an appointee of President George W. Bush who also worked in Mr. Bush’s White House, clerked at the Supreme Court for Justice Kennedy. He was also a prosecutor under the independent counsel Kenneth W. Starr, who investigated President Bill Clinton.

Mr. Trump has expressed a desire to name a woman to the court, and Judge Barrett is a favorite of religious conservatives. Deeply religious and a former law clerk for Justice Scalia, she once argued that Catholic judges should sometimes recuse themselves from sentencing in death penalty cases.

Judge Kethledge also clerked for Justice Kennedy, and has the support of some conservative activists. A graduate of the University of Michigan Law School, Judge Kethledge does not have the Harvard or Yale pedigree that Mr. Trump has told associates he would like to see in the next justice.

Mr. Trump said all of the judges he had talked to about the job were “outstanding people,” but he gave no hint about whom he might choose. “They are really incredible people in so many different ways — academically, and every other way,” the president said.

“I had a very, very interesting morning,” he said, speaking to reporters in the Oval Office before a meeting with Prime Minister Mark Rutte of the Netherlands.

Mr. Trump said he most likely would meet with two or three other candidates before making his decision. A list of the names of those interviewed on Monday was earlier reported by the Above the Law blog and The Washington Post.

Sarah Huckabee Sanders, the White House press secretary, said Mr. Trump did not meet with any other candidates over the weekend. With less than a week before Mr. Trump’s self-imposed deadline of next Monday to announce his choice to succeed Justice Kennedy, who announced his resignation on Wednesday, the White House has embarked on a vetting and public relations effort that in past administrations has taken weeks.

But Ms. Sanders said that teams of lawyers from the counsel’s office and the Justice Department have begun compiling volumes of research on the finalists, and that key staff members would spend the summer focused entirely on winning confirmation of whomever Mr. Trump eventually chooses. And Democrats have responded to the accelerated White House effort by immediately targeting judges widely believed to be on the president’s shortlist.

The candidates interviewed Monday are among a group of federal appeals court judges who are believed to be finalists to replace Justice Kennedy. All are on a broader list of 25 people, mostly conservative judges, from which Mr. Trump has publicly said he will choose. His shortlist appears to also include Thomas M. Hardiman of the Third Circuit, William H. Pryor Jr. of the 11th Circuit and Joan L. Larsen of the Sixth Circuit.

Even without a name, Democrats began focusing on Monday on the backgrounds and rulings of the finalists, stressing their likely opposition to the Affordable Care Act and predicting that any of them would support overturning Roe v. Wade, the landmark 1973 case establishing the constitutional right to an abortion.

Senator Chuck Schumer of New York, the Democratic leader, took direct aim at Ms. Barrett on Twitter, accusing her of supporting the idea of rearguing the abortion precedent.

“The bottom line: Judge Barrett has given every indication that she will be an activist judge on the Court,” Mr. Schumer wrote in one of a series of tweets. “If chosen as the nominee, she will be the deciding vote to overturn Roe v. Wade and to strike down pre-existing conditions protections in the ACA.”

Mr. Schumer and other Democrats have insisted that Mr. Trump’s choice for the court must be pressed during hearings to specifically say whether they would join a majority in the court to end abortion rights.

But Senator Mitch McConnell, Republican of Kentucky and the majority leader, responded to that demand on Monday by highlighting quotes from liberal justices saying during their confirmation hearings that it would be inappropriate to talk about specific cases.

Mr. McConnell said such comments amounted to “the Ginsburg Standard,” named after Justice Ruth Bader Ginsburg.

“A judge sworn to decide impartially can offer no forecasts, no hints, for that would show not only disregard for the specifics of the particular case, it would display disdain for the entire judicial process,” Justice Ginsburg said during her 1993 hearings to be on the court. She added, “Were I to rehearse here what I would say and how I would reason on such questions, I would act injudiciously.”

Democratic strategists are also eager to highlight Mr. Trump’s statements about abortion during the 2016 presidential campaign. In one debate, Mr. Trump said he would appoint two or three anti-abortion justices to the court, so a more conservative majority could overturn Roe v. Wade.

“That will happen automatically in my opinion,” Mr. Trump said during the debate with Hillary Clinton, his Democratic opponent, in October 2016.

“I would have walked out the door,” Judge Gorsuch said. “That’s not what judges do.”

At the White House on Monday, Ms. Sanders, the press secretary, repeatedly said Mr. Trump had not asked candidates for the court to say how they would rule on specific cases. “He’s looking for individuals that have the right intellect, the right temperament and that will uphold the Constitution,” Ms. Sanders told reporters.

In the meantime, White House officials said, Mr. Trump will temporarily reorganize his White House staff to focus on confirming a new justice by the time the court’s next term opens in October.

Raj Shah, a deputy press secretary, will take a leave from his responsibilities in the press office to focus exclusively on coordinating the president’s message on behalf of the pick, Ms. Sanders said.

Donald F. McGahn II, the White House counsel, will lead the process, Ms. Sanders said, and will be aided by a team of lawyers in the counsel’s office and another at the Justice Department, which will help in vetting the candidates and preparing the nominee for hearings.

The job of working with conservative organizations outside the White House will fall to Justin Clark, the director of the Office of Public Liaison, Ms. Sanders said.

“Teams of attorneys from the White House Counsel’s Office and Department of Justice are working to ensure the president has all the information he needs to choose his nominee,” Ms. Sanders said in a statement. “The Department of Justice is fully engaged to support the nomination and confirmation efforts.”

The staff deployment is a reflection of the seriousness with which the White House takes the task of winning a quick confirmation. While Republicans control the Senate, they have only a one-vote margin, and Mr. McConnell has made it clear he wants to bring the nomination to a vote before the November congressional elections.

Others in the White House, including Marc Short, the director of legislative affairs, and John F. Kelly, the chief of staff, will also be involved in the process, Ms. Sanders said.